Yıl 2023,
, 84 - 104, 30.03.2023
Coşkun Yakar
,
Hazm Talab
Kaynakça
- Arslan, M., Yakar, C., Terminal value problems with causal operators, HJMS, 48(5) (2018), 897-907. https://doi.org/10.15672/HJMS.2018.566
- Bhaskar, T. G., Devi, J. V., Nonuniform stability and boundedness criteria for set differential equations, Applicable Analysis, 84(2) (2005), 131-143. https://doi.org/10.1080/00036810410001724346
- Bhaskar, T. G., Devi, J. V., Stability criteria for set differential equations, Mathematical and Computer Modelling, 41(11-12) (2005), 1371-1378.
- Brauer, F., Nohel, J. A., The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover, NY, USA, 1989.
- Chadaram, A. N., Dhaigude, D. B., Devi, J .V., Stability results in terms of two measures for set differential equations involving causal operators, European Journal of Pure and Applied Mathematics, 10(4) (2017), 645-654.
- Çiçek, M., Yakar, C., Oğur, B., Stability, boundedness, and Lagrange stability of fractional differential equations with initial time difference, The Scientific World Journal, 2014, Article ID 939027 (2014), 1-7. https://doi.org/10.1155/2014/939027
- Corduneanu, C., Functional Equations with Causal Operators, CRC Press, 2002. https://doi.org/10.1201/9780203166376
- Devi, J. V., Existence, uniqueness of solutions for set differential equations involving causal operators with memory, EJPAM, 3(4) (2010), 737-747.
- Devi, J. V., Comparison theorems and existence results for set differential equations involving causal operators with memory, Nonlinear Studies, 18(4) (2011), 603-610.
- Devi, J. V., Generalized monotone iterative technique for set differential equations involving causal operators with memory, International Journal of Advances in Engineering Sciences and Applied Mathematics, 3(1-4) (2011), 74-83. https://doi.org/10.1007/s12572-011-0031-1
- Devi, J. V., Chadaram A. N., Boundedness results for impulsive set differential equations involving causal operators with memory, Communications in Applied Analysis, 17(1) (2013), 9-19.
- Devi, J. V., Chadaram A. N., Stability results for impulsive set differential equations involving causal operators with memory, Global Journal of Mathematical Sciences: Theory & Practical, 2(2) (2014), 49-53.
- Devi, J. V., Chadaram A. N., Stability results for set differential equations involving causal operators with memory, European Journal of Pure and Applied Mathematics, 5(2) (2012), 187-196.
- Drici, Z., Mcrae, F. A., Devi, J. V., Stability results for set differential equations with causal maps, Dynamic Systems and Applications, 15(3) (2006), 451-464.
- Gücen, M. B., Yakar, C., Strict stability of fuzzy differential equations by Lyapunov functions, International Scholarly and Scientific Research & Innovation, 12(5) (2018), 315-319. https://doi.org/10.5281/ZENODO.1316718
- Lakshmikantham, V., On the stability and boundedness of differential systems, Math. Proc. Camb. Phil. Soc., 58(3) (1962), 492-496. https://doi.org/10.1017/S030500410003677X
- Lakshmikantham, V., Leela, S., Martynyuk, A. A., Practical Stability of Nonlinear Systems, World Scientific, Singapore, 1991.
- Lakshmikantham, V., Leela, S., Vatsala, A. S., Setvalued hybrid differential equations and stability in terms of two measures, International Journal of Hybrid Systems, 2(2) (2002), 169-188.
- Lakshmikantham, V., Leela, S., Martynyuk, A. A., Stability Analysis of Nonlinear Systems, NY, USA: M. Dekker, 1989.
- Lakshmikantham, V., Leela, S., Devi, J. V., Stability theory for set differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 11(2-3) (2004), 181-190.
- Lakshmikantham, V., Leela, S., Drici, Z., McRae, F. A., Theory of Causal Differential Equations, Atlantis Studies in Mathematics for Engineering and Science, 2010. https://doi.org/10.2991/978-94-91216-25-1
- Lakshmikantham, V., Bainov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. https://doi.org/10.1142/0906
- Lakshmikantham, V., Bhaskar, T. G., Devi, J. V., Theory of Set Differential Equations in Metric Spaces, Cottenham, Cambridge, Cambridge Scientific Publishers, 2006.
- Lakshmikantham, V., Matrosov, V. M., Sivasundaram, S., Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Dordrecht, Boston, USA, Kluwer Academic Publishers, 1991.
- Lakshmikantham, V., Deo, S. G., Method of Variation of Parameters for Dynamic Systems, Amsterdam, Netherlands, Gordon and Breach Science Publishers, 1998.
- Lakshmikantham, V., Leela, S., Differential and Integral Inequalities: Theory and Applications, New York, USA, Academic Press, 1969.
- Lakshmikantham, V., Liu, X. Z., Stability Analysis in Terms of Two Measures, World Scientific, Singapore, 1993. https://doi.org/10.1142/2018
- Lakshmikantham, V., Rama Mohana Rao, M., Theory of Integro-Differential Equations, Lausanne, Switzerland, Gordon and Breach Science Publishers, 1995.
- Lakshmikantham, V., Vatsala, A. S., Differential inequalities with initial time difference and applications, Journal of Inequalities and Applications, 3(3) (1999), 233-244. https://doi.org/10.1155/S1025583499000156
- Lakshmikantham, V., Vatsala, A. S., Theory of Differential and Integral Inequalities with Initial Time Difference and Applications, In: Rassias, T.M. and Srivastava, H.M. (eds.) Analytic and Geometric Inequalities and Applications, Vol 478, Dordrecht, Netherlands, Springer, 1999. https://doi.org/10.1007/978-94-011-4577-0 12
- LaSalle, J., Lefschetz, S., Stability by Liapunov’s Direct Methods with Applications, Mathematics in Science and Engineering, New York, Academic Press, 1961.
- Liu, X., Shaw, M. D., Boundedness in terms of two measures for perturbed systems by generalized variation of parameters, Communications in Applied Analysis, 5(4) (2001), 435-444.
- Shaw, M. D., Yakar, C., Generalized variation of parameters with initial time difference and a comparison result in terms of Lyapunov-like functions, International Journal of Nonlinear Differential Equations Theory-Methods and Applications, 5(1-2) (1999), 86-108.
- Shaw, M. D., Yakar, C., Stability criteria and slowly growing motions with initial time difference, Problems of Nonlinear Analysis in Engineering Systems, 6 (2000), 50-66.
- Smith, H., An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-7646-8
- Tu, N. N., Tung, T. T., Stability of set differential equations and applications, Nonlinear Analysis: Theory, Methods & Applications, 71(5-6) (2009), 1526-1533. https://doi.org/10.1016/j.na.2008.12.045
- Yakar, C., Çiçek, M., Gücen, M. B., Boundedness and Lagrange stability of fractional order perturbed system related to unperturbed systems with initial time difference in Caputo’s sense, Advances in Difference Equations, 54 (2011), 1-14. https://doi.org/10.1186/1687-1847-2011-54
- Yakar, C., Boundedness Criteria in Terms of Two Measures with Initial Time Difference, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, Watam Press, Waterloo, 2007, 270-275.
- Yakar, C., Bal., B., Yakar, A., Monotone technique in terms of two monotone functions in finite system, Journal of Concrete and Applicable Mathematics, 9(3) (2011), 233-239.
- Yakar, C., Çiçek, M., Gücen, M. B., Practical stability, boundedness criteria and Lagrange stability of fuzzy differential systems, Computers & Mathematics with Applications, 64(6) (2012), 2118-2127. https://doi.org/10.1016/j.camwa.2012.04.008
- Yakar, C., Ciçek, M., Initial time difference boundedness criteria and Lagrange stability, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 18(6) (2011), 797-811.
- Yakar, C., Çiçek, M., Theory, methods and applications of initial time difference, boundedness and Lagrange stability in terms of two measures for nonlinear systems, Hacettepe Journal of Mathematics and Statistics, 40(2) (2011), 305-330.
- Yakar, C., Gücen, M. B., Initial time difference stability of causal differential systems in terms of Lyapunov functions and Lyapunov functionals, Journal of Applied Mathematics, 2014, Article ID 832015, (2014), 1-7. https://doi.org/10.1155/2014/832015
- Yakar, C., Shaw, M. D., A comparison result and Lyapunov stability criteria with initial time difference, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 12(6) (2005), 731-737.
- Yakar, C., Shaw, M. D., Initial time difference stability in terms of two measures and variational comparison result, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 15(3) (2008), 417-425.
- Yakar, C., Shaw, M. D., Practical stability in terms of two measures with initial time difference, Nonlinear Analysis: Theory, Methods & Applications, 71(12) (2009), e781–e785. https://doi.org/10.1016/j.na.2008.11.039
- Yakar, C., Talab, H., Stability of perturbed set differential equations involving causal operators in regard to their unperturbed ones considering difference in initial conditions, Advances in Mathematical Physics, 2021, Article ID 9794959, (2021), 1-12. https://doi.org/10.1155/2021/9794959
Lagrange stability in terms of two measures with initial time difference for set differential equations involving causal operators
Yıl 2023,
, 84 - 104, 30.03.2023
Coşkun Yakar
,
Hazm Talab
Öz
In this paper, we investigate generalized variational comparison results aimed to study the stability properties in terms of two measures for solutions of Set Differential Equations (SDEs) involving causal operators, taking into consideration the difference in initial conditions. Next, we employ these comparison results in proving the theorems that give sufficient conditions for equi-boundedness, equi-attractiveness in the large, and Lagrange stability in terms of two measures with initial time difference for the solutions of perturbed SDEs involving causal operators in regard to their unperturbed ones.
Kaynakça
- Arslan, M., Yakar, C., Terminal value problems with causal operators, HJMS, 48(5) (2018), 897-907. https://doi.org/10.15672/HJMS.2018.566
- Bhaskar, T. G., Devi, J. V., Nonuniform stability and boundedness criteria for set differential equations, Applicable Analysis, 84(2) (2005), 131-143. https://doi.org/10.1080/00036810410001724346
- Bhaskar, T. G., Devi, J. V., Stability criteria for set differential equations, Mathematical and Computer Modelling, 41(11-12) (2005), 1371-1378.
- Brauer, F., Nohel, J. A., The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover, NY, USA, 1989.
- Chadaram, A. N., Dhaigude, D. B., Devi, J .V., Stability results in terms of two measures for set differential equations involving causal operators, European Journal of Pure and Applied Mathematics, 10(4) (2017), 645-654.
- Çiçek, M., Yakar, C., Oğur, B., Stability, boundedness, and Lagrange stability of fractional differential equations with initial time difference, The Scientific World Journal, 2014, Article ID 939027 (2014), 1-7. https://doi.org/10.1155/2014/939027
- Corduneanu, C., Functional Equations with Causal Operators, CRC Press, 2002. https://doi.org/10.1201/9780203166376
- Devi, J. V., Existence, uniqueness of solutions for set differential equations involving causal operators with memory, EJPAM, 3(4) (2010), 737-747.
- Devi, J. V., Comparison theorems and existence results for set differential equations involving causal operators with memory, Nonlinear Studies, 18(4) (2011), 603-610.
- Devi, J. V., Generalized monotone iterative technique for set differential equations involving causal operators with memory, International Journal of Advances in Engineering Sciences and Applied Mathematics, 3(1-4) (2011), 74-83. https://doi.org/10.1007/s12572-011-0031-1
- Devi, J. V., Chadaram A. N., Boundedness results for impulsive set differential equations involving causal operators with memory, Communications in Applied Analysis, 17(1) (2013), 9-19.
- Devi, J. V., Chadaram A. N., Stability results for impulsive set differential equations involving causal operators with memory, Global Journal of Mathematical Sciences: Theory & Practical, 2(2) (2014), 49-53.
- Devi, J. V., Chadaram A. N., Stability results for set differential equations involving causal operators with memory, European Journal of Pure and Applied Mathematics, 5(2) (2012), 187-196.
- Drici, Z., Mcrae, F. A., Devi, J. V., Stability results for set differential equations with causal maps, Dynamic Systems and Applications, 15(3) (2006), 451-464.
- Gücen, M. B., Yakar, C., Strict stability of fuzzy differential equations by Lyapunov functions, International Scholarly and Scientific Research & Innovation, 12(5) (2018), 315-319. https://doi.org/10.5281/ZENODO.1316718
- Lakshmikantham, V., On the stability and boundedness of differential systems, Math. Proc. Camb. Phil. Soc., 58(3) (1962), 492-496. https://doi.org/10.1017/S030500410003677X
- Lakshmikantham, V., Leela, S., Martynyuk, A. A., Practical Stability of Nonlinear Systems, World Scientific, Singapore, 1991.
- Lakshmikantham, V., Leela, S., Vatsala, A. S., Setvalued hybrid differential equations and stability in terms of two measures, International Journal of Hybrid Systems, 2(2) (2002), 169-188.
- Lakshmikantham, V., Leela, S., Martynyuk, A. A., Stability Analysis of Nonlinear Systems, NY, USA: M. Dekker, 1989.
- Lakshmikantham, V., Leela, S., Devi, J. V., Stability theory for set differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 11(2-3) (2004), 181-190.
- Lakshmikantham, V., Leela, S., Drici, Z., McRae, F. A., Theory of Causal Differential Equations, Atlantis Studies in Mathematics for Engineering and Science, 2010. https://doi.org/10.2991/978-94-91216-25-1
- Lakshmikantham, V., Bainov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. https://doi.org/10.1142/0906
- Lakshmikantham, V., Bhaskar, T. G., Devi, J. V., Theory of Set Differential Equations in Metric Spaces, Cottenham, Cambridge, Cambridge Scientific Publishers, 2006.
- Lakshmikantham, V., Matrosov, V. M., Sivasundaram, S., Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Dordrecht, Boston, USA, Kluwer Academic Publishers, 1991.
- Lakshmikantham, V., Deo, S. G., Method of Variation of Parameters for Dynamic Systems, Amsterdam, Netherlands, Gordon and Breach Science Publishers, 1998.
- Lakshmikantham, V., Leela, S., Differential and Integral Inequalities: Theory and Applications, New York, USA, Academic Press, 1969.
- Lakshmikantham, V., Liu, X. Z., Stability Analysis in Terms of Two Measures, World Scientific, Singapore, 1993. https://doi.org/10.1142/2018
- Lakshmikantham, V., Rama Mohana Rao, M., Theory of Integro-Differential Equations, Lausanne, Switzerland, Gordon and Breach Science Publishers, 1995.
- Lakshmikantham, V., Vatsala, A. S., Differential inequalities with initial time difference and applications, Journal of Inequalities and Applications, 3(3) (1999), 233-244. https://doi.org/10.1155/S1025583499000156
- Lakshmikantham, V., Vatsala, A. S., Theory of Differential and Integral Inequalities with Initial Time Difference and Applications, In: Rassias, T.M. and Srivastava, H.M. (eds.) Analytic and Geometric Inequalities and Applications, Vol 478, Dordrecht, Netherlands, Springer, 1999. https://doi.org/10.1007/978-94-011-4577-0 12
- LaSalle, J., Lefschetz, S., Stability by Liapunov’s Direct Methods with Applications, Mathematics in Science and Engineering, New York, Academic Press, 1961.
- Liu, X., Shaw, M. D., Boundedness in terms of two measures for perturbed systems by generalized variation of parameters, Communications in Applied Analysis, 5(4) (2001), 435-444.
- Shaw, M. D., Yakar, C., Generalized variation of parameters with initial time difference and a comparison result in terms of Lyapunov-like functions, International Journal of Nonlinear Differential Equations Theory-Methods and Applications, 5(1-2) (1999), 86-108.
- Shaw, M. D., Yakar, C., Stability criteria and slowly growing motions with initial time difference, Problems of Nonlinear Analysis in Engineering Systems, 6 (2000), 50-66.
- Smith, H., An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-7646-8
- Tu, N. N., Tung, T. T., Stability of set differential equations and applications, Nonlinear Analysis: Theory, Methods & Applications, 71(5-6) (2009), 1526-1533. https://doi.org/10.1016/j.na.2008.12.045
- Yakar, C., Çiçek, M., Gücen, M. B., Boundedness and Lagrange stability of fractional order perturbed system related to unperturbed systems with initial time difference in Caputo’s sense, Advances in Difference Equations, 54 (2011), 1-14. https://doi.org/10.1186/1687-1847-2011-54
- Yakar, C., Boundedness Criteria in Terms of Two Measures with Initial Time Difference, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, Watam Press, Waterloo, 2007, 270-275.
- Yakar, C., Bal., B., Yakar, A., Monotone technique in terms of two monotone functions in finite system, Journal of Concrete and Applicable Mathematics, 9(3) (2011), 233-239.
- Yakar, C., Çiçek, M., Gücen, M. B., Practical stability, boundedness criteria and Lagrange stability of fuzzy differential systems, Computers & Mathematics with Applications, 64(6) (2012), 2118-2127. https://doi.org/10.1016/j.camwa.2012.04.008
- Yakar, C., Ciçek, M., Initial time difference boundedness criteria and Lagrange stability, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 18(6) (2011), 797-811.
- Yakar, C., Çiçek, M., Theory, methods and applications of initial time difference, boundedness and Lagrange stability in terms of two measures for nonlinear systems, Hacettepe Journal of Mathematics and Statistics, 40(2) (2011), 305-330.
- Yakar, C., Gücen, M. B., Initial time difference stability of causal differential systems in terms of Lyapunov functions and Lyapunov functionals, Journal of Applied Mathematics, 2014, Article ID 832015, (2014), 1-7. https://doi.org/10.1155/2014/832015
- Yakar, C., Shaw, M. D., A comparison result and Lyapunov stability criteria with initial time difference, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 12(6) (2005), 731-737.
- Yakar, C., Shaw, M. D., Initial time difference stability in terms of two measures and variational comparison result, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 15(3) (2008), 417-425.
- Yakar, C., Shaw, M. D., Practical stability in terms of two measures with initial time difference, Nonlinear Analysis: Theory, Methods & Applications, 71(12) (2009), e781–e785. https://doi.org/10.1016/j.na.2008.11.039
- Yakar, C., Talab, H., Stability of perturbed set differential equations involving causal operators in regard to their unperturbed ones considering difference in initial conditions, Advances in Mathematical Physics, 2021, Article ID 9794959, (2021), 1-12. https://doi.org/10.1155/2021/9794959